3. Let G be a group and let SG be the set of all permutations on G. For each g € G, define Kg: GG as Kg(a) = gag ¹, for all a € G. (a) (b) (c) (d) -Prove that Kg is a group automorphism of G. Note: An automorphism of G is an isomorphism from G to G. Show that I(G):= {Kg | g € G} is a subgroup of SG- Let Z= {g € G | ga = ag, for all a € G}. Prove that ZAG and that G/ZI(G). Prove that if G/Z is cyclic, then G is abelian.

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